1. Field of the Invention
The present invention relates to an apparatus and a method that corrects multi-phase signals for detecting a position of an object and obtains a phase corresponding to the position of the object. The present invention is applicable to, for example, an encoder, a laser interferometer or the like.
2. Description of the Related Art
After converting such multi-phase signals (analog signals) such as two-phase signals into digital signals by an A/D converter, performing arc-tangent calculation thereon enables acquisition of minute position information. AD converters include ones having a resolution of 12 to 18 bits, and use of such AD converters enables acquisition of position information corresponding to one several thousandth to one several hundred thousandth of one period of the two-phase signals. However, the position information thus acquired contains (a) an error caused due to a distortion component included in the two-phase signals on the basis of which the position information is acquired and (b) drift in a signal processing circuit.
U.S. Pat. No. 4,458,322 discloses a method for removing an error contained in two-phase signals. This method measures an offset error and an amplitude error (that is, mismatched correlation) by using a maximum value and a minimum value of each of the two-phase signals and then corrects the errors.
Moreover, U.S. Pat. No. 5,581,488 discloses a method for detecting and correcting errors other than the offset error (zeroth-order error component) and the amplitude error (first-order error component) by using maximum and minimum values of each of a sum signal and a difference signal of the two-phase signals in addition to the maximum and minimum values of each of the two-phase signals. This method measures and corrects a phase difference error (an error from π/2 that is a phase difference between phases of the two-phase signals), a second-order distortion and a third-order distortion in addition to the offset error and the amplitude error.
Error components contained in a position signal include low-order error components and high-order error components which become non-negligible with subdivision of signals by digital processing. However, the conventional error measuring method using the maximum and minimum values of the two-phase signals corresponds to error measurement on the basis of information on a radius (Lissajous radius) of a figure (Lissajous figure or (Lissajous waveform) drawn as a trajectory of an intersection of the two-phase signals when the two-phase signals are allocated to respective axes of orthogonal coordinates. This method cannot measure the high-order error components in principle.
Specifically, Fourier expansion of the two-phase signals (x, y) can express these signals as the following expressions where a symbol Σ represents a sum from k equals zero to infinity of values in brackets:x=cos θ+Σ(ak cos kθ+dk sin kθ)y=sin θ+Σ(ek cos kθ+bk sin kθ).
Moreover, a variation Rk of the Lissajous radius due to kth-order error components (ak, dk, ek, bk) in A and B phase signals (two-phase signals) is shown by the following expression:Rk=0.5(ak+bk)cos(k−1)θ+0.5(dk−ek)sin(k−1)θ+0.5(ak−bk)cos(k+1)θ+0.5(dk+ek)sin(k+1)θ.
As clear from the above expression, the kth-order error component included in the two-phase signals causes (k+1)th-order and (k−1)th-order periodic error components of the Lissajous radius. Therefore, analysis of the variation of the Lissajous radius cannot distinguish whether that variation is caused due to an effect of the (k+1)th-order error component or the (k−1)th-order error component, which makes it impossible to specify error factors included in the position signal.
A reason that the conventional error measuring method using the maximum and minimum values of the two-phase signals is available is that the zeroth-order error component and the first-order error component have a special property which other order error components do not have.
Specifically, the method is available on the zeroth-order error component (offset error) because it includes no sine component, and a (k−1)th-order (that is, minus first-order) variation component of the Lissajous radius caused due to the zeroth-order error component appears by aliasing as a first-order variation component to overlap a (k+1)th-order variation component. However, when an original signal includes a second-order error component, this (k−1)th-order variation component appears as the first-order variation component of the Lissajous radius which cannot be separated. Therefore, this method is available when the second-order error component is negligible.
Furthermore, all variation components of the Lissajous radius caused due to the first-order error component (amplitude error) appears on a (k+1)th-order side since the amplitude errors a1 and b1 have a restricted relationship of a1=−b1 and the phase difference errors d1 and e1 have a restricted relationship of e1=d1.
In more detail, an influence of the error component given to a phase error (or Lissajous radius error) is expressed by the following expression showing an inner product of an error vector and a tangential unit vector (or radial unit vector):(ak cos kθ+dk sin kθ,ek cos kθ+bk sin kθ)·(−sin θ, cos θ).
Multiplying the kth-order error component by a sine function generates plus and minus first-order error components.
Therefore, it becomes possible to specify the first-order error component contained in the original signal from the second-order variation component of the Lissajous radius. However, also in this case, the (k−1)th-order variation component due to a third-order error component contained in the original signal appears as the second-order variation component of the Lissajous radius which cannot be separated.
The measurements of the zeroth-order and first-order error components using the Lissajous radius are performed based on a premise that the original signal does not contain second- or higher-order error components. The method disclosed in U.S. Pat. No. 5,581,488 predetermines second- and third-order error components based on a presumption that a distortion of a signal processing circuit contains only an error component whose peak coincides with that of a fundamental wave component, enabling measurements of the respective order error components. However, this presumption is only approximately true, which is insufficient to detect and correct the entire error components included in the two-phase signals.
Japanese Patent No. 1933273 discloses a method for detecting and correcting error components by Fourier transform of each of two-phase signals. This disclosed method measures a phase of a signal from a measurement apparatus in a state where the measurement apparatus is fixed at a predetermined angle by using an angle setter such as a micrometer, to calculate the error components. Such a configuration is not suitable for automatic measurement of errors of the measurement apparatus during actual use thereof.
Japanese Patent Laid-Open No. 2003-0254785 discloses a method of calculating, based on a fact that phase information calculated from two-phase signals linearly varies with movement of a scale at a constant speed, a deviation from the line as an error. However, since Japanese Patent Laid-Open No. 2003-0254785 does not disclose that error correction is performed with a general expression (polynomial expression) focusing on coefficients of high-order distortion components, and only discloses that extraction of error correction data and correction are performed separately from each other, the error may be corrected insufficiently.
In order to perform more highly accurate error correction, it is firstly necessary to calculate coefficients of high-order error components to enable the error correction using a general expression (polynomial expression), and secondly necessary to provide an algorithm enabling continuous and successive calculation of the coefficients of the high-order error components and correction thereof.
Applying the method disclosed in Japanese Patent No. 1933273 which performs the Fourier transform on each of the two-phase signals enables independent measurement of amplitude coefficients of high-order errors, which seems to be able to be used as it is for a purpose of automatic measurement and correction of the errors during actual use of the apparatus.
However, in this case, how phase information used for the Fourier transform is obtained is a problem.
In general, it is desirable that error components contained in the two-phase signals be very few (for example, 1% or less of signal amplitude). Therefore, phase information obtained by performing arc-tangent calculation on the two-phase signals containing the error components seems to be able to be used for this purpose.
However, the phase information thus obtained contains a phase error caused due to the error component contained in the input two-phase signals (error of a result value of the arc-tangent calculation). As a result, the Fourier transform is performed with the phase error.
That is, although Fourier transform is performed by multiplying a signal by sin kθ and cos kθ and then integrating the product thereof over one period in principle, an error contained in θ (phase) generates an error in a process of the Fourier transform.
A synergetic effect of fundamental wave components (a1, b1, a2, b2, a3, b3, . . . ) of the two-phase signals and phase error involvement components (d1, e1, d2, e2, d3, e3, . . . ) causes a false (spurious) error component of the phase θ in a result of the Fourier transform.
A detailed description will hereinafter be made of this phenomenon.
The kth-order error component causes a phase error δk expressed by the following expression (characters are same as those in the above description). The following expression shows that a kth-order distortion component of the two-phase signals becomes (k−1)th- and (k+1)th-order periodic error components.δk=((ak+bk)/2)sin(k−1)θ+((ek−dk)/2)cos(k−1)θ+((bk−ak)/2)sin(k+1)θ+((dk+ek)/2)cos(k+1)θ
When the phase contains the error δk, an extracted signal includes an error corresponding to a product of a differential coefficient for the phase of the signal and −δk. Spurious errors resulting therefrom are expressed as follows when approximation is made with neglecting square or higher terms of the error coefficient.
The error overlapping a cosine signal is expressed by the following expression:δk sin θ=−(ak/2)cos kθ−(dk/2)sin kθ+((ak+bk)/4)cos(k−2)θ−((ek−dk)/4)sin(k−2)θ+((ak−bk)/4)cos(k+2)θ+((ek+dk)/4)sin(k+2)θ.
The error overlapping a sine signal is expressed by the following expression:−δk cos θ=−(ek/2)cos kθ−(bk/2)sin kθ−((ek−dk)/4)cos(k−2)θ−((ak+bk)/4)sin(k−2)θ−((ek+dk)/4)cos(k+2)θ+((ak−bk)/4)sin(k+2)θ.
That is, the phase error caused due to the error component contained in the two-phase signals compresses the originally existing error component into ½, and further generates a false error having a size of ¼ of the originally existing error component and being away therefrom by plus and minus second orders.
Table 1 and Table 2 show sizes of the false errors (spurious errors) generated due to the error components (ak, dk, ek, bk) of the respective orders. Table 1 shows ak and bk, and Table 2 shows dk and ek. Table 1 considers only the error components ak and bk relating to the amplitude error, and Table 2 considers only the error components dk and ek relating to the phase error. However, neither ak nor bk of the error components contained in the original signal do not influence dk and ek to be measured, and neither dk nor ek of the error components contained in the original signal do not influence ak and bk to be measured.
Since amounts of false error components with respect to those of true error components are known, predetermined inverse matrices of Table 1 and Table 2 enable inverse calculation of the error components contained in the original signal from the measured error components. However, a matrix generating the false error component is a singular matrix, and therefore it has no inverse matrix. Accordingly, it is impossible to analytically calculate, by using the periodic error of the Lissajous waveform, the kth-order distortion components (ak, dk, ek, bk) where k is for example from 0 to 6.